Torus Knots in Adjoint Representation

TL;DR

This work derives a closed-form expression for the adjoint HOMFLY-PT polynomial of torus knots $T[m,n]$ by applying the Rosso-Jones formula with Adams operations in the $A_{N-1}$ (adjoint) sector. The authors exploit the simple plethystic expansion of the adjoint representation into composite representations to obtain a compact, explicit sum over hook-shaped representations, yielding the main result $H^{T[m,n]}_{Adj}(q,A)=\frac{A^{2mn}}{qD_{Adj}(q)}\left(m-1+\sum_{a,b=1}^m(-1)^{a+b}A^{-2n}q^{-2n(a+b-m-1)}\cdot qD_{([a,1^{m-a}],[b,1^{m-b}])}(q,A)\right)$. The paper also analyzes symmetry properties, specializations (Alexander, special polynomial, colored Jones), and the stable homology limit, illustrating the role of adjoint invariants in Vogel's universality and potential extensions to links and other groups, as well as refined theories. The results provide a concrete, general tool for adjoint knot invariants in torus knots and connect to broader structures in topological quantum field theory and representation theory.

Abstract

We derive a closed-form expression for the adjoint polynomials of torus knots and investigate their special properties. The results are presented in the very explicit double sum form and provide a deeper insight into the structure of adjoint invariants essential for the Vogel's universality of Chern-Simons theory.